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Inégalités fonctionnelles liées aux formes de Dirichlet. De l'isopérimétrie aux inégalités de Sobolev.

Abstract : Ergodic Markov semigroups provide approximations of probability measures by means of some functional inequalities. The aim of the thesis is to investigate some of these inequalities, from isoperimetry to Sobolev inequalities. What we are essentially interested in is to establish some links between these inequalities, to determine their optimal constants and to get some criteria which ensure their existence. Three main issues were investigated. Firstly we examine the links between logarithmic Sobolev inequality (LS) and Bobkov Gaussian isoperimetric inequality (BGI). It is shown that semigroups whose curvature is bounded from below (by a possibly negative number) and which satisfy (LS) also satisfy a (BGI) inequality. We hence get a (BGI) inequality for some spins systems. In the second part, we prove that the exact order of the Poincaré constant for a log-concave probability measure on the real line is given by the square of the mean value of the distance to the median. The proof is based on a variation computation in the set of convex functions. The final part of the work is devoted to establishing new criteria for Sobolev inequalities when the Bakry-Emery curvature-dimension (CD) criterion fails. The way we use is based on the construction (by means of conformal changes of metric and tensorization) of a Dirichlet structure in a higher dimension which satisfy a (CD) inequality and projects onto the initial structure in the lower dimension.
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Contributor : Pierre Fougères <>
Submitted on : Tuesday, March 25, 2003 - 2:45:09 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 8:10:17 PM


  • HAL Id : tel-00002624, version 1


Pierre Fougères. Inégalités fonctionnelles liées aux formes de Dirichlet. De l'isopérimétrie aux inégalités de Sobolev.. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2002. Français. ⟨tel-00002624⟩



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